To find the standard form of the equation of a hyperbola given its foci and vertices, we will first identify the key characteristics of the hyperbola based on the information provided.
The foci of the hyperbola are at (0, 8) and (0, -8), while the vertices are at (0, 6) and (0, -6). This indicates that the hyperbola is oriented vertically since the foci and vertices are aligned along the y-axis.
The general form of the equation for a hyperbola that opens vertically is:
(y – k)²/a² – (x – h)²/b² = 1
Where (h, k) is the center of the hyperbola, ‘a’ is the distance from the center to each vertex, and ‘b’ relates to the distance from the center to the foci.
1. **Finding the Center:**
The center of the hyperbola is the midpoint between the two foci or the two vertices. Using the vertices, the center can be calculated as:
Center (h, k) = (0, ((6) + (-6))/2) = (0, 0)
2. **Finding ‘a’:**
The value of ‘a’ is the distance from the center to each vertex:
a = 6 – 0 = 6
(Thus, a² = 6² = 36)
3. **Finding ‘c’:**
The distance from the center to each focus is denoted as ‘c’:
c = 8 – 0 = 8
(Thus, c² = 8² = 64)
4. **Finding ‘b’:**
We can find ‘b’ using the relationship between a, b, and c for hyperbolas:
c² = a² + b²
64 = 36 + b²
b² = 64 – 36 = 28
5. **Writing the Equation:**
Now we have all the values needed:
- h = 0
- k = 0
- a² = 36
- b² = 28
Plugging these values into the equation format, we get:
(y – 0)²/36 – (x – 0)²/28 = 1
6. **Final Standard Form:**
The standard form of the equation for the hyperbola in question is:
y²/36 – x²/28 = 1
This equation defines the hyperbola with the given foci and vertices, accurately reflecting its geometric properties.